This document discusses using virtual manipulatives to teach fractions to students with learning disabilities. It begins by noting that students with learning disabilities often struggle with fractions and their conceptual understanding. Recent research supports using interactive visual models or virtual manipulatives to teach fractions as they allow for adjustable instruction, scaffolding of content, and increased practice opportunities. The document then discusses trends in using technology to teach math to students with disabilities. It provides examples of virtual manipulative tools and their features. Benefits of using virtual manipulatives include helping students follow visual images, providing individualized accommodations, and allowing active student engagement. Potential challenges include technical difficulties and ensuring students understand the connection between the virtual manipulatives and mathematical concepts.

1. Intervention in School and Clinic 2017, Vol. 52(3) 148 –153
© Hammill Institute on Disabilities 2016
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DOI: 10.1177/1053451216644830
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Feature Article
Many students with learning disabilities (LD) demonstrate
poor conceptual understanding of fractions, limiting their
ability to solve more advanced problem-solving tasks,
including ratios, rates, and proportions (Siegler et al., 2010).
When working with students with LD who have individual-
ized education programs in the area of mathematics, recent
research supports the use of interactive visual models called
virtual manipulatives (Bouck & Flanagan, 2010; Shin,
2013) because of the ability to enable the design features to
adjust instructional demands, scaffold mathematical con-
tent, and increase the amount of practice opportunities, all
of which can be incorporated into intensified instruction for
students with LD (B. R. Bryant et al., 2016).
Initially, virtual manipulatives were defined as web-
based images on a computer monitor that permitted students
to manipulate a visual model as if it were three-dimensional
(Bouck & Flanagan, 2010). Previous syntheses and meta-
analyses of methods of teaching fractions to students with
LD recommended the use of visual models (e.g., Shin &
D. P. Bryant, 2015) because of their potential in helping stu-
dents to better understand the meaning and relationship of

2. fractions through relevant diagrams (e.g., part-whole rela-
tionship is represented with a divided circle, multiplication
of fractions is represented with a rectangular area model).
Through technology, students can access virtual manipula-
tives on mobile devices such as iPads in addition to web-
based computer applications. This capability to easily
access virtual manipulatives allows students opportunities
to engage with the visual models as part of their interven-
tion on various concepts and skills related to fractions. For
example, teachers can use a variety of virtual visual models
such as fraction bars, area models, and number lines to
teach equivalent fractions and addition, subtraction, multi-
plication, and division of fractions. This article discusses (a)
technology trends in teaching mathematics to students with
LD, (b) virtual manipulatives as instructional mathematical
tools for use in the classroom, (c) the benefits of using
644830 ISCXXX10.1177/1053451216644830Intervention in
School and ClinicShin et al.
research-article2016
1Ewha Womans University, Seoul, South Korea
2University of Texas at Austin, The Meadows Center for
Preventing
Educational Risk, Austin, TX, USA
3University of Massachusetts, Lowell, MA, USA
4IDEA Public Schools, Austin, TX, USA
5University of Hawaii at Manoa, Honolulu, HI, USA
Corresponding Author:
Mikyung Shin, Ewha Womans University, Eonju-ro 157-gil, 22,
Gangnam-
gu, Seoul 135-895, South Korea.
Email: [email protected]
Virtual Manipulatives: Tools for Teaching Mathematics

3. to Students With Learning Disabilities
Mikyung Shin, PhD1, Diane P. Bryant, PhD2, Brian R. Bryant,
PhD2,
John W. McKenna, PhD3, Fangjuan Hou, PhD4, and Min Wook
Ok, PhD5
Abstract
Many students with learning disabilities demonstrate difficulty
in developing a conceptual understanding of mathematical
topics. Researchers recommend using visual models to support
student learning of the concepts and skills necessary to
complete abstract and symbolic mathematical problems. Virtual
manipulatives (i.e., interactive visual models) can be used
as tools for students while actively engaging in learning
mathematics. This article discusses (a) technology trends in
teaching
mathematics to students with learning disabilities, (b) virtual
manipulatives as instructional mathematical tools for use in
the classroom, (c) the benefits of using virtual manipulatives,
and (d) potential challenges with using virtual manipulatives
for instructional purposes.
Keywords
learning disabilities, mathematics intervention, middle school
students, virtual manipulatives, visual models
Shin et al. 149
virtual manipulatives, and (d) potential challenges with
using virtual manipulatives for instructional purposes.
Technology Trends

4. Over the past 30 years, computers have been used in special
education for independent games, drill practices, and tutori-
als for improving mathematics knowledge of and profi-
ciency with skills and concepts such as basic mathematics
facts, computation, and problem solving (Seo & D. P. Bryant,
2009). Federal policy and standards (e.g., National
Governors Association Center for Best Practices & Council
of Chief State School Officers [NGAC/CCSSO], 2010) have
emphasized the importance of infusing technology into
mathematics practice as a means for promoting mathemati-
cal understanding. Moreover, in the digital world, schools
and teachers are expected to provide engaging opportunities
for students with disabilities through the use of integrating
technology into instructional practice (Edyburn, 2013).
As a result of this emphasis on integrating technology
into mathematics instruction, most of the research on vir-
tual manipulatives as an instructional tool has focused on
typical students in general education settings. For exam-
ple, Moyer-Packenham and Westenskow (2013) con-
ducted a meta-analysis of virtual manipulatives. They
found 32 studies that showed a moderate effect of virtual
manipulatives on math performance compared to typical
instruction in the general education classroom (e.g.,
instruction with physical manipulatives and/or abstract
mathematics symbols but not the use of virtual manipula-
tives as a tool). Recently, other researchers have investi-
gated the effects of virtual manipulatives as a tool with
students with LD using a single-subject research design.
For example, Reneau (2012) and Shin (2013) investigated
the effects of using virtual manipulatives as a tool for
teaching fraction concepts and skills within the context of
word problem solving to students with LD. The findings
from these two studies showed improvements in word
problem-solving, which included problems with fractions,
during the intervention phase compared to performance in

5. the baseline condition. In another study, Satsangi and
Bouck (2015) found virtual manipulatives an effective
tool for teaching the concept of area and perimeter to stu-
dents with LD. This effect was maintained over time and
generalized to abstract word problems. Based on recent
research on the use of virtual manipulatives as tools for
teaching mathematics to students with LD, a number of
potential resources are available to classroom teachers.
Virtual Manipulatives as Instructional
Mathematical Tools
In responding to the needs of “web natives” who are adept
at accessing and manipulating technology devices (Brown,
2013, p. 55), virtual manipulatives provide a variety of
classroom opportunities. Teachers can implement virtual
manipulatives as a form of interactive visual models to
teach the concept of fractions during their lessons. When
virtual manipulatives are presented in the form of games
and quizzes, teachers can use them to check for understand-
ing of fraction concepts and skills. Table 1 summarizes the
features of web-based and iPad applications of virtual
manipulatives for selected products.
Teachers can integrate the use of visual manipulative
tools into their instruction by demonstrating how the mod-
els can be used to teach new concepts, such as fractions,
and providing students opportunities to engage in interac-
tive activities during guided and/or independent practice.
Several web-based virtual manipulatives applications pro-
vide build your own fractions options, whereby teachers
can create their own fraction problems and control virtual
manipulatives as visual model tools for instruction.
Because virtual manipulatives provide a visual model tool,
teachers must be able to guide students in how to use them

6. and make the connection between visual models and what
they represent (Moyer-Packenham & Westenskow, 2013).
The Common Core State Standards for Mathematics
(NGAC/CCSSO, 2010) include the recommendation for
using various tools, such as virtual manipulatives. This rec-
ommendation can be translated into instruction by having
students manipulate visual models and importantly explain
their models in relation to mathematical equations to be
solved. For example, when solving fraction problems,
teachers should demonstrate how to manipulate quantities
Table 1. Summary of the Features of Virtual Manipulatives.
Name and URL
Web
App
iPad
App
Visual
Model
Corrective
Feedback Audio
Video
Tutorial
Build
Fractions
Conceptua Math http://www.conceptuamath.com Yes Yes
Multiple Yes Yes Yes Yes
Illuminations http://illuminations.nctm.org/Games-Puzzles.aspx
Yes Yes Multiple Yes No No Yes

7. NLVM http://nlvm.usu.edu/en/nav/vlibrary.html Yes No
Multiple Yes No No Yes
Fun Fraction http://funfraction.org Yes No Area Yes Yes Yes
No
Virtual Manipulatives! http://ABCya.com No Yes Multiple No
No No Yes
Dividing Fractions https://www.brainingcamp.com/content/
dividing-fractions/
Yes Yes Multiple No No yes Yes
http://www.conceptuamath.com
http://illuminations.nctm.org/Games-Puzzles.aspx
http://nlvm.usu.edu/en/nav/vlibrary.html
http://funfraction.org
http://ABCya.com
https://www.brainingcamp.com/content/dividing-fractions/
https://www.brainingcamp.com/content/dividing-fractions/
150 Intervention in School and Clinic 52(3)
that illustrate the meaning of the fraction number (i.e.,
numerator and denominator quantities). Students should be
expected to talk about how they are using the visual model
tools to show their work while solving problems.
Free and available web-based applications offer opportuni-
ties for teachers and students to implement virtual manipula-
tives (i.e., visual model tools) during mathematics instruction.
For example, using Illuminations (http://illuminations.nctm.
org/Games-Puzzles.aspx) or Conceptua Math (http://www.
conceptuamath.com), teachers can provide instruction in frac-
tions during whole-class and small-group instruction. These
web-based applications offer various visual model tools (e.g.,
pattern block, number line, vertical bar, dots) for each relevant

8. fraction concept (e.g., equivalent fractions, addition, subtrac-
tion, multiplication, division, word problems). Using interac-
tive whiteboards, teachers can demonstrate how to connect
virtual manipulatives to each fraction concept. In addition,
using iPad applications (e.g., ABCya.com’s Virtual
Manipulatives!, Brainingcamp’s Dividing Fractions), teach-
ers can help individual students explore fractions using their
own mobile devices. After the teacher has introduced con-
cepts and skills related to equivalent fractions or dividing frac-
tions, students can individually explore fraction visual models
using iPad apps. Students can drag pieces of circles or fraction
bars into the workspaces. By comparing the size of sliced
circles or length of fractions bars, students can compare
equivalent fractions. In particular, Virtual Manipulatives! pro-
vides note functions through which students can draw any
other fraction diagrams and write equations if needed.
Design Features of Virtual
Manipulatives
One design feature of virtual manipulative tools (e.g., National
Library of Virtual Manipulatives [http://nlvm.usu.edu/en/nav/
vlibrary.html], Fun Fraction [http://funfraction.org]) and iPad
applications (e.g., Conceptua Math’s Student App,
Brainingcamp’s Dividing Fractions) is the ability to monitor
student progress with specific concepts and skills through the
use of virtual-manipulatives-based quizzes and games (Bouck
& Flanagan, 2010; Edyburn, 2013). In this student-centered
learning and evaluation environment, virtual-manipulatives-
based quizzes and games provide corrective feedback after
learning trials. Through the hints and prompts provided by the
applications, students can check their answers. Also, by using
virtual manipulatives, teachers can determine their students’
ability to manipulate base-10 blocks, graph equations, and
determine parts for multiplying or dividing fractions to repre-
sent problems and find solutions.

9. Students also can be provided with individualized scaf-
folds through several design features in virtual-manipula-
tives-based quizzes and games. For example, certain models
(e.g., fraction bars, circles, rectangular area models, number
line bars) provide multiple visual models to represent
fraction concepts and skills (e.g., equivalent fractions,
multiplication, division). Thus, students can easily focus on
previously constructed visual models to solve their fraction
problems. Some applications provide multiple visual models
for each application so that students can select either circle
or square models. When solving word problems with frac-
tions, students can listen to the question through read-aloud
functions in Conceptua Math and Fun Fraction. In addition,
when students continually fail to solve word problems and
forget how to use virtual manipulatives for learning frac-
tions, students can review lessons by watching videos in the
applications (e.g., Fun Fraction).
Figure 1 describes how a student can create and apply
virtual manipulatives (i.e., Fun Fraction) to solve mathe-
matical word problems such as this: “There are 2
3
feet of
ribbon for making bows. I used 1
2
of the ribbon. How many
feet of ribbon did I use to make bows?” Fun Fraction ini-
tially includes four mathematical problem-solving steps:
read, restate, represent, and Answer. Of the four steps,

10. screenshots in Figure 1 show the represent step, which
includes virtual manipulatives for representing the problem
situation. In solving mathematical word problem using the
virtual rectangular area model in Fun Fraction, a student
can go through nine instructional steps.
Step 1: Identify two quantities: vertical line (used rib-
bon), horizontal line (original ribbon).
Step 2: Because there are 2
3
feet of original ribbon, divide
1 unit into 3 pieces (denominator of the multipli-
cand) by increasing the denominator arrow but-
ton to 3.
Step 3: Shade 2
3
of the original ribbon by shifting the green
slider to the right.
Step 4: Because I used 1
2
of the ribbon, divide 1 unit into
2 pieces (denominator of the multiplier) by
increasing the denominator arrow button to 2.
Step 5: Shade 1
2

11. of the used ribbon by shifting the yellow
slider to up.
Step 6: Count the number of sections in 1 unit square.
That is the denominator of the product.
Step 7: Count the number of purple color sections. That
is the numerator of the product.
Step 8: Thus, 1
2
of 2
3
is 2
6
.
Step 9: Namely, 1
2
× 2
3
= 2
6
.
In this way, the student can use virtual manipulatives to
solve the word problem and connect the virtual visual model
to numeric symbols to find the solution.

12. Benefits of Virtual Manipulatives
There are several major benefits for special education
teachers to use virtual manipulatives in the classroom when
http://illuminations.nctm.org/Games-Puzzles.aspx
http://illuminations.nctm.org/Games-Puzzles.aspx
http://www.conceptuamath.com
http://www.conceptuamath.com
http://nlvm.usu.edu/en/nav/vlibrary.html
http://nlvm.usu.edu/en/nav/vlibrary.html
http://funfraction.org
Shin et al. 151
teaching students with LD. First, virtual manipulatives help
students follow their virtual mental images on the screen
using built in visual and numeric models. Suh and Moyer
(2008) indicated, “The applet seemed to benefit special needs
learners by giving them built in supports for the mathemati-
cal ideas that reduced their cognitive overload” (p. 303). By
using the computer-built-in pictorial images and symbolic
notations, students could be free to focus on the mathematical
connections and relationships (Moyer-Packenham, Salkind,
& Bolyard, 2008; Suh & Moyer, 2008). Second, the applica-
tion of virtual manipulatives acts as individualized accom-
modations for students with LD and mathematics difficulties
(D. P. Bryant & B. R. Bryant, 2011; Edyburn, 2013). That is,
students can control the learning process by adjusting their
own pace and repeating the practice, if necessary (Claes, Van
Hove, Vandevelde, van Loon, & Schalock, 2012). Third, the
use of virtual manipulatives tools allows students to actively
engage in their learning (Satsangi & Bouck, 2015). For

13. Question: There are 2
3
feet of ribbon for making bows. I used 1
2
of the ribbon. How many feet of ribbon did I use to make
bows?
Step1: Identify two quantities: vertical line (used ribbon),
horizontal line
(original ribbon)
Step 4: Because I used 1
2
of the ribbon, divide 1 unit into 2 pieces (denomi-
nator of the multiplier) by increasing the denominator arrow
button to 2.
Step 2: Because there are 2
3
feet of original ribbon, divide 1 unit into 3
pieces (denominator of the multiplicand) by increasing the
denominator
arrow button to 3.
Step 5: Shade 1
2
of the used ribbon by shifting the yellow slider to up.
Step 3: Shade 2

14. 3
of the original ribbon by shifting the green slider to the
right.
Step 6: Count the number of sections in 1 unit square. That is
the denomi-
nator of the product.
Step 7: Count the number of purple color sections. That is the
numerator
of the product.
Step 8: Thus, 1
2
of 2
3
is 2
6
Step 9: Namely, 1
2
× 2
3
= 2
6
Figure 1. Results of a student completing prompts in the
Represent step. This figure illustrates the procedural steps of
how to use
virtual manipulatives. Used by permission from
http://funfraction.org.

15. http://funfraction.org
152 Intervention in School and Clinic 52(3)
example, recent studies have demonstrated that virtual
manipulatives encourage active engagement of students with
LD and increase their academic achievement in fraction con-
cepts and skills (Reneau, 2012; Shin, 2013). Fourth, many
virtual manipulative websites are available free of charge and
allow easy access to everyone (Bouck & Flanagan, 2010).
Teachers with busy class schedules can access these ready-
to-use online resources efficiently. Teachers do not need to
clean up and store manipulatives after using the visuals, thus
their time can be better devoted to planning and implement-
ing instruction for their students with LD.
Challenges and
Solution
s With Virtual
Manipulatives
The use of virtual manipulatives for teaching fractions with
students with LD presents four major challenges. First, in
general, most teachers have not been trained on how to use
and select the right device to meet individual students’
needs (McMahon & Walker, 2014). To alleviate the chal-

16. lenge of incorporating virtual manipulatives into their
instruction, teachers can use video tutorials and instruction
provided in virtual manipulative websites and applications.
See Table 1 for websites and applications that provide video
tutorial options. Using the tutorials and instruction, teachers
can easily follow how to implement the virtual manipula-
tives in their lesson plans. However, only some of the web-
sites and applications provide the tutorials; virtual
manipulatives websites and applications developers should
consider providing the tutorials for users.
Second, although the video tutorials can be useful for
teachers, they are insufficient. Many researchers have
warned that virtual manipulatives in and of themselves do
not guarantee that students are developing mathematical
conceptual understanding (Moyer-Packenham &
Westenskow, 2013). When using virtual manipulatives,
teachers must check students’ ability to use virtual manipu-
latives to accurately connect their visual models to mathe-
matics concepts (McMahon & Walker, 2014). During this
process, students should be encouraged to verbalize their
mathematical thinking and justify their representations of
problem situations (Hunt, 2014). Teachers should monitor
students’ actions of linking two dynamic visual models on
the screen or touch pad by asking reflective questions such

17. as these: “What does the virtual fraction bar represent in the
given situation? What happened to the graph when you
changed the spinner? How can we define the mathematical
relationship between the two virtual area models on the
screen?” (Hunt, 2014; Moyer-Packenham & Westenskow,
2013). In this way, students should be shown a visual repre-
sentation of a mathematical concept and asked to explain it;
or, in the reverse situation, given problems such as equiva-
lent fractions, students should be able to represent this
concept using virtual manipulatives (Dougherty, D. P.
Bryant, B. R. Bryant, Darrough, & Pfannenstiel, 2015).
Third, teachers must employ thoughtful selection cri-
teria to choose from a vast array of visual models offered
on websites and as applications. Much like software,
there is a need to consider instructional design features
particularly as they relate to the instructional needs of stu-
dents with LD. For instance, teachers can utilize a rubric
when evaluating and selecting web-based or iPad applica-
tions. Ok, Kim, Kang, and B. R. Bryant (2016) developed
a rubric form that teachers can use to evaluate instruc-
tional applications for teaching students with LD. The
rubric form includes three sections: (a) identifying infor-
mation such as content area, objectives, and type of appli-

18. cation, (b) evaluating instructional features such as
examples, progress monitoring, and feedback, and (c)
grading options. Using the rubric can be helpful for teach-
ers to identify and select high-quality virtual manipula-
tives applications.
Finally, in recognizing the need for intensive instruc-
tion for students with LD, teachers should embed the use
of virtual visual models during guided practice of mathe-
matical concepts and skills as part of “intensive, strategic,
and explicit” instruction (B. R. Bryant et al., 2016, p. 9).
Moreover, because not all students respond to virtual
visual models in the same way, teachers will likely need to
modify the instructional delivery process. For example,
teachers can scaffold instruction by first providing physi-
cal manipulatives and then applying virtual manipulatives
in the extension of the previously taught concepts (B. R.
Bryant et al., 2016; Leh & Jitendra, 2012). Instead of using
all of the multimedia options provided by virtual manipu-
latives websites and applications, teachers can carefully
select mathematical inputs and symbols from the multiple
options to incorporate as a means for intensifying instruc-
tion (Mayer & Moreno, 2003).
Concluding Thoughts

19. By promoting an interactive learning environment, teachers
can help students with LD engage in their learning of math-
ematics. The use of virtual manipulatives is a viable tool for
teaching fraction concepts and skills to students with LD.
Virtual manipulatives can be integrated into teacher-deliv-
ered instruction as a visual model tool to facilitate students’
conceptual understanding. Also, they can be used as a form
of checking for understanding or as games and quizzes dur-
ing differentiated instruction or remediation (Regan,
Berkeley, Hughes, & Kirby, 2014). Regardless of the
instructional situation, teachers must monitor students to
make sure they fully understand what they are doing with
the tools and, as a result, have a meaningful learning
experience.
Shin et al. 153
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect
to the research, authorship, and/or publication of this article.

20. Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
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